Following the guest post from Tom Banks on challenges to eternal inflation, we’re happy to post a follow-up to this discussion by Don Page. Don was a graduate student of Stephen Hawking’s, and is now a professor at the University of Alberta. We have even collaborated in the past, but don’t hold that against him.

Don’s reply focuses less on details of eternal inflation and more on the general issue of how we should think about quantum gravity in a cosmological context, especially when it comes to counting the number of states. Don is (as he mentions below) an Evangelical Christian, but by no means a Young Earth Creationist!

Same rules apply as before: this is a technical discussion, which you are welcome to skip if it’s not your cup of tea.

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I tend to agree with Tom’s point that “it is extremely plausible, given the Bekenstein Hawking entropy formula for black holes, that the quantum theory of a space-time , which is dS in both the remote past and remote future, has a finite dimensional Hilbert space,” at least for four-dimensional spacetimes (excluding issues raised by Raphael Bousso, Oliver DeWolfe, and Robert Myers for higher dimensions in Unbounded entropy in space-times with positive cosmological constant) if the cosmological constant has a fixed finite value, or if there are a finite number of possible values that are all positive. The “conceptual error … that de Sitter (dS) space is a system with an ever increasing number of quantum degrees of freedom” seems to me to arise from considering perturbations of de Sitter when it is large (on a large compact Cauchy surface) that would evolve to a big bang or big crunch when the Cauchy surface gets small and hence would prevent the spacetime from having both a remote past and a remote future. As Tom nicely puts it, “In the remote past or future we can look at small amplitude wave packets. However, as we approach the neck of dS space, the wave packets are pushed together. If we put too much information into the space in the remote past, then the packets will collide and form a black hole whose horizon is larger than the neck. The actual solution is singular and does not resemble dS space in the future.”

So it seems to me that, for fixed positive cosmological constant, we can have an arbitrarily large number of quantum states if we allow big bangs or big crunches, but if we restrict to nonsingular spacetimes that expand forever in both the past and future, then the number of states may be limited by the value of the cosmological constant.

This reminds me of the 1995 paper by Gary Horowitz and Robert Myers, The value of singularities, which argued that the timelike naked singularity of the negative-mass Schwarzschild solution is important to be excluded in order to eliminate such states which would lead to energy unbounded below and instabilities from the presumably possible production (conserving energy) of arbitrarily many possible combinations of positive and negative energy. Perhaps in a similar way, big bang and big crunch singularities are important to be excluded, as they also would seem to allow infinitely many states with positive cosmological constant.

Now presumably we would want quantum gravity states to include the formation and evaporation of black holes (or of what phenomenologically appear similar to black holes, whether or not they actually have the causal structure of classical black holes), which in a classical approximation have singularities inside them, so presumably such `singularities’ should be allowed, even if timelike naked singularities and, I would suggest, big bang and big crunch singularities should be excluded. Perhaps one can postulate that one should restrict to states of immortal de Sitter spacetime, which has no timelike naked singularities anywhere, and which is asymptotically a single region in the very distant past that is locally de Sitter (though globally it can be highly distorted) and which is also asymptotically a single region in the very distant future that is likewise locally de Sitter, without any big crunch or big bang singularities in between, or between more than one asymptotic region (as one would expect to get from perturbations of the Nariai metric that lead to asymptotic de Sitter regions separated by big bang and big crunch singularities). Such singularities might be considered mortal wounds for de Sitter, allowing an infinite number of states to fester up from such wounds, killing the hope for a finite number of states and for unitarity. On the other hand, a localized black hole that forms within de Sitter could be considered a wound that is not mortal and which can be healed by Hawking evaporation without going outside the assumed finite number of quantum states for immortal de Sitter with a strictly positive cosmological constant. I have considered such a possibility in my paper No-Bang Quantum State of the Cosmos.

On the other hand, even if we find such a restriction for positive cosmological constant to a finite set of states, the number of such states for the observed tiny value of the cosmological constant seems so huge that it would seem surprising to me if most of them can explain our observations, so I now am skeptical of the maximally mixed state I proposed in No-Bang Quantum State of the Cosmos and now tend to prefer a pure state like what I proposed in Symmetric-Bounce Quantum State of the Universe. When coupled with a solution to the measure problem, such as what I proposed in Agnesi Weighting for the Measure Problem of Cosmology, this pure state seems to be consistent with all our observations, though of course these are highly preliminary proposals and are not yet fully precisely specified, nor are they nearly so simple and esthetically pleasing as I would hope for in a final complete theory of the universe. We would like a simple quantum state for the universe and simple rules for extracting the probabilities of observations from it. (In The Born Rule Dies and in related papers, I showed that the Born rule, interpreted mathematically in the form that probabilities of observations are expectation values of projection operators, does not work in a universe large enough for multiple copies of an observation, but the probabilities still could be expectation values of other quantum operators, though it is so far unknown what they should be; this knowledge would be a solution of the measure problem if the quantum state were also known.)

Some people seem to prefer the simple state that is the maximally mixed state out of a finite-dimensional Hilbert space, but if this does not explain our observations, I see no objection to postulating that the state is some other simple state, perhaps (but not necessarily) a pure state. By Occam’s razor, we scientists tend to ascribe higher prior probabilities to simpler theories, so we would tend to prefer a simpler quantum state, but I don’t think we should take so narrow a view that such a simple state has to be a thermal state, or a maximally mixed state.

Now even if the quantum state of the universe is chosen (perhaps as a simple mixed state, perhaps as a simple pure state) from a finite set of states, or even if it is a particular pure state, I’m agnostic as to whether or not this could be a state with eternal inflation. Eternal inflation may lead to a huge universe in which there are an arbitrarily large number of nearby states, but those may be states with mortal perturbations (evolving forward and/or backward in time to mortal singularities in a classical description) that would be excluded from the allowed states. So I don’t see that there is any problem with having disallowed states that at some time look nearby to the allowed states. Our universe looks very much as if it could have had a big bang in its past (which would allow an infinite number of states), and presumably a generic quasiclassical perturbation of the present state would evolve back to such a mortal singularity, but the actual state of our universe might have been one of the very large but finite number of states that did not have such a mortal singularity but instead had a bounce. If it did have a bounce, the size at the bounce seems that it might have been much smaller than the throat of de Sitter with the observed value of the cosmological constant, which suggests to me that the quantum state is much more restricted than just the restriction to the large but finite number of states that have bounces rather than mortal singularities.

The question seems to be open, even if the state is immortal de Sitter (by which I mean having a positive cosmological constant but no mortal singularities, not that it is metrically de Sitter or even close to de Sitter), whether this state consists of a superposition of quasiclassical spacetimes with most of them with significant amplitudes having a huge or infinite number of bubbles that keep forming and branching off (and presumably attaching on as well). Even from the considerations of Tom Banks and of my discussion above, I don’t see any obvious reason why it might not. One could presumably have a single simple pure state such that, if it were decomposed into quasiclassical components, would have components that are individually very complex, rather as the binary representation of the Bible, or of the Library of Congress, or of the eprint arXiv, or the entire Internet, or of all the words that have ever been written on earth, is each a very large and presumably very complex integer but is just one component of the set of all integers, which is a simple whole, of which almost all of the separate parts are individually much more complex.

I would guess that presumably at least some quasiclassical components of any simple quantum state that could describe our universe would be very complex and perhaps have a huge number of bubble or pocket subuniverses that could be described as having eternal inflation. Perhaps a more relevant question is whether the probabilities of our observations are given by something like a path integral that is dominated by such eternal inflating spacetimes, or whether the path integral (or whatever procedure gives the quantum probabilities) is dominated by simpler spacetimes, such as ones with only one region and a single bounce. For thermodynamic reasons I presently have some slight personal inclination toward the latter view, that for the dominant contribution it is sufficient to consider only one bounce in the past, at roughly five trillion days ago (an easily memorized value for the age of the universe that fits to four digits the middle of the range of the current measurements of 13.69(13) Gyr), rather than an infinite sequence of bubble formations in my past. Though I am not a Young Earth Creationist as some of my fellow Evangelical Christians are, perhaps I am still a Young Universe Creationist (for thermodynamic reasons rather than for theological reasons, since I personally do not see any theological reason that God could not have created an infinite eternally inflating universe) in the sense that I suspect that most of the quantum amplitude for our observations can be ascribed to quasiclassical universes (or spacetimes in a path integral) that each had its smallest spatial size (say in a foliation of Cauchy surfaces that are closed hypersurfaces of constant extrinsic curvature, to exclude nearly null hypersurfaces of arbitrarily tiny three-volume) not much more than about five trillion days ago. But so far as I can see, this question of whether or not eternally inflating spacetimes are important for the quantum probabilities of our observations is very much an open question, even if the set of quantum states is restricted to be finite or to be a single pure state.